Articulated, closed kinematic chain planar monopod

ABSTRACT

The present invention concerns a novel leg mechanism for quadrupedal locomotion. This design engages a linkage to couple assembly that only requires a single degree of actuation. The topological arrangement of the system produces a foot trajectory that is well-suited for dynamic gaits including trot-running, bounding, and galloping.

RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/562,149 filed Sep. 22, 2017, and herein incorporated by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH & DEVELOPMENT

This invention was made with government support National ScienceFoundation under Grant No. 1557312. The government has certain rights inthe invention.

INCORPORATION BY REFERENCE OF MATERIAL SUBMITTED ON A COMPACT DISC

Not applicable.

BACKGROUND OF THE INVENTION

The transition of legged robots from test-benches into real-worldscenarios becomes viable only when the dynamic locomotion and maneuversare developed enough to require only high-level inputs to operate thesecomplex machines. A significant amount of research by the scientificcommunity is motivated by this hypothesis, including the exploration ofhybrid dynamic control frameworks. Hybrid dynamics can be defined as acomposition of stance and flight domains that alternate when triggeredby instantaneous impacts. In addition to the challenge of utilizinghybrid dynamics, popular approaches towards the design of legged robotsdictate the inclusion of multiple degrees of freedom (DOF) per leg whichnecessitates multi-layered controllers that further increase complexity.Generally, to take a single step, these machines have a whole-bodytrajectory generator at the higher level. At the lower level, suchsystems have (i) a foot trajectory generator that is guided by the bodytrajectory (ii) an inverse kinematic controller that plans the legmotion with respect to the generated foot trajectory while avoidingsingularities. Despite impressive performance in laboratory settings,active research is aimed at building robust controllers to account fordelays and singularities before legged robots may navigate urbanenvironments. An alternative approach that has shown promise inquadrupedal platforms is the implementation of only one actuator perleg. The articulated nature of legged robots is most effectivelycaptured by a closed kinematic chain (CKC) mechanism due to the abilityto control passive degrees of freedom with a single actuator throughclosed chains of linkages. Additional advantages of CKC mechanisms, forthe purposes of legged robotics, are reduced weight due to theconcentration of actuators at a proximal location and an increasedrigidity-to-weight ratio. These properties are of great value inhigh-speed applications such as dynamic locomotive gaits. While thehigher-level control functionalities remain complex due to the hybridnature of dynamic legged locomotion, an indirect advantage of a singleDOF CKC mechanism is that it mechanically encodes robustness at thelower level by directly eliminating the need for the previouslyhighlighted foot trajectory generators and on-board inverse kinematiccalculations. Given that the majority of existing one DOF mechanisms areat best quasi-statically stable systems, a single DOF CKC mechanism hasbeen used whose topological arrangement ensures a trajectory thatpromotes dynamic locomotion.

The ramifications of this design choice are revealed during an effort totransform the dynamic model into state space form. Principally, the CKCsare characterized by algebraic equations (AE) and the resultant systemsof equations that describe the system are identified as differentialalgebraic equations (DAEs). From a simulation standpoint, numericalsolutions of DAEs are more challenging to obtain in comparison toordinary differential equations (ODEs). Within robotics, constrainedmechanisms are defined by index-3 DAEs. The index represents the numberof times that holonomic constraints must be differentiated with respectto time before the form of ODE can be assumed. One of the existingmethods in literature proposes direct interaction with index-3 DAEsthrough input-output linearization. Another technique suggestsdifferentiation of holonomic constraints twice, thus representing themat the velocity level, and then solving the AE to obtain an implicitstate space representation of the resultant index-1 DAE. However, aby-product of this method is the magnification of drift in solution.Furthermore, the admissibility of the result is solely dependent on thesatisfaction of the initial condition. Drift stabilization formulationshave been proposed in the past to address this issue. Amongst these,Baumgarte's stabilization method is a widely adopted scheme. Yet itsappeal is shadowed by the difficulty of choosing appropriate parametersto guarantee robustness. Moreover, from the control perspective a richlibrary of stable model-based controllers exists for dynamicsrepresented by ODEs in explicit state space form but are not readilyextended to DAE descriptions that are implicit in nature.

Beyond conventional practices of dealing with DAEs directly, singularperturbation formulation (SPF) avoids the limitations by approximatingthe DAE as an ODE. They were first implemented on the model of atwo-phase flow heat exchanger to express the DAEs in explicit statespace form. This method was adapted to a fixed base CKC robot where theAE is substituted by an asymptotically stable ODE that characterizes theconstraint violation. The resultant ODE is also known as the fastdynamics ODE. The success of this approach lies in the rapiddisappearance of this fast dynamics term, thus resulting in aconvergence to the slower subsystem. It is noteworthy that the SPFtreatment results in second-order ODEs that are equal in number to theindependent generalized coordinates that describe the system.

BRIEF SUMMARY OF THE INVENTION

In one embodiment, the present invention provides a method, approach,and solution that concern a legged robot with a novel topologicalarrangement, that includes two closed loops which may be kinematicloops. The loops may include a vertically oriented floating-base, withone active joint and one passive revolute joint (hip joint) arranged,parallel to the frontal plane and having a driving link connected to theactive joint and the hip flexion and extension link connected to theother end of the driving link with another revolute joint. Also providedis a knee flexion and extension link mounted on a revolute joint locatedon the hip flexion and extension link, close to its point of attachmentwith the driving link.

A femur link may be mounted on the passive revolute joint, located onthe base with two other revolute joints, one close to its point ofattachment with the base and one at the other end on which the tibialink is mounted. The free end of the hip flexion and extension link ismounted on the first revolute joint of the femur link, forming the firstclosed loop. The tibia link has a revolute joint, located close to itsconnecting point with the femur link, which connects the free end of theknee flexion and extension link, forming the second closed loop. Thedistal end of the tibia joint is mounted with a compression spring ofpredetermined stiffness value to provide compliance during impact.

In other embodiments, the present invention provides a legged robotwherein the lower end of the spring is fitted with a compliant rubberfoot to provide a second-stage of compliance against impact forces.

In other embodiments, the present invention provides a legged robotwherein the flight-phase trajectory of the foot is jerk free and has aretraction rate that reduces energy losses at touchdown.

In other embodiments, the present invention provides a legged robotwherein the stance-phase trajectory is a sinusoidal curve, thatestablishes contact with the ground at its lowest point generatingground-reaction forces enough to perform dynamic gaits.

In other embodiments, the present invention provides a legged robot withone continuously rotating actuator to perform dynamic locomotion (bound,trot, amble and canter).

In other embodiments, the present invention provides a legged robot witha novel topological arrangement, with two closed loops, comprising avertically oriented floating-base, with one active joint and one passiverevolute joint (hip joint) arranged, parallel to the frontal plane andhaving a driving link connected to the active joint and the hip flexionand extension link connected to the other end of the driving link withanother revolute joint; a knee flexion and extension link mounted on arevolute joint located on the hip flexion and extension link, close toits point of attachment with the driving link; a femur link mounted onthe passive revolute joint, located on the base with two other revolutejoints, one close to its point of attachment with the base and one atthe other end on which the tibia link is mounted; the free end of thehip flexion and extension link is mounted on the first revolute joint ofthe femur link, forming the first closed loop; the tibia link has arevolute joint, located close to its connecting point with the femurlink, which connects the free end of the knee flexion and extensionlink, forming the second closed loop; the tibia link being a rigid linkwith no compliance elements (spring); and a compliance controller thatimplements a virtual spring acting between the active joint on the baseand the foot.

In other embodiments, the present invention provides a legged robotwherein the compliance can be varied based on the required clearancebetween the foot and the ground.

In other embodiments, the present invention provides a dynamicquadrupedal robot comprised of four legs; each leg comprising of: avertically oriented floating-base, with one active joint and one passiverevolute joint (hip joint) arranged, parallel to the frontal plane andhaving a driving link connected to the active joint and the hip flexionand extension link connected to the other end of the driving link withanother revolute joint; a knee flexion and extension link mounted on arevolute joint located on the hip flexion and extension link, close toits point of attachment with the driving link; a femur link mounted onthe passive revolute joint, located on the base with two other revolutejoints, one close to its point of attachment with the base and one atthe other end on which the tibia link is mounted; the free end of thehip flexion and extension link is mounted on the first revolute joint ofthe femur link, forming the first closed loop; the tibia link has arevolute joint, located close to its connecting point with the femurlink, which connects the free end of the knee flexion and extensionlink, forming the second closed loop. The tibia link being a rigid linkwith no compliance elements (spring); and a compliance controller thatimplements a virtual spring acting between the active joint on the saidbase and the foot.

In other embodiments, the present invention provides a quadrupedal robotwherein the front legs and back legs are mounted on two differentsagittal planes creating an offset for enhanced stability.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory onlyand are not restrictive of the invention, as claimed.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

In the drawings, which are not necessarily drawn to scale, like numeralsmay describe substantially similar components throughout the severalviews. Like numerals having different letter suffixes may representdifferent instances of substantially similar components. The drawingsgenerally illustrate, by way of example, but not by way of limitation, adetailed description of certain embodiments discussed in the presentdocument.

FIG. 1 shows (a) design and dimension specifications for an embodimentof the present invention. Left: Frontal view. (i) driving disk (ii) HFElink (iii) femur (iv) KFE link (v) tibia (vi) double compliance (springand rubber foot tip) (vii) absolute encoder. (b) Close-up view of thetwo kinematic loops present in the mechanism. (c) Angular relationshipssimplify the structure so that it can be described as an OKC forkinematic purposes.

FIGS. 2A-2B show (a) notations used to formulate the optimizationproblem, the critical pivots and their respective constraint evolutionspace and (b) designed ideal trajectory utilized in the optimization.

FIG. 3 is a comparison of the optimized angular profiles of loop oneagainst the profiles obtained from heuristic link lengths in onecomplete rotation.

FIGS. 4A-4B show a comparison of the optimized trajectories of the knee(a) and the foot (b) against the profiles obtained from heuristic linklengths in one complete rotation. The heuristic foot trajectory is outof scale and not pictured.

FIG. 5 is a state machine implementation of the stance and flight phasecontroller.

FIG. 6 illustrates the virtual separation method aided to alleviatemodeling of the CKC for a robot of the present invention on the left.Towards the right, rejoining of the separated joints and reduction ofgeneralized coordinates that completely describe the robot is depicted.

FIGS. 7A, 7B and 7C show (a) measurement conventions utilized to derivethe dynamic model of an embodiment of the present invention in theflight phase, illustrated as an unconstrained system. (b) Relevant linklengths that describe the first closed loop of the system. (c) Depictthe constraint definitions employed in the formulation, e0=e1, and g0=g1and link lengths pertinent to the second loop.

FIG. 8 illustrates the singularly perturbed dynamic model.

FIG. 9 is a simulation sequence of an embodiment of the presentinvention's flight phase while performing a running motion at 3.2 m/s.

FIG. 10 shows the asymptotic tracking of constraint error to zero.Constraints are invariant to the hybrid dynamic framework.

FIGS. 11A-11B show (a) Horizontal and vertical position of an embodimentof the present invention. A constant angle of attack assures uniform legheight (y). (b) Limit cycles of PRESENT INVENTION running for 12 stepsin the simulator. Pictured on the left is the phase portrait of theheight and on the right is the evolution of the driving joint.

FIG. 12 illustrates a commanded angle of the driving joint (q₁) overtime.

FIG. 13 illustrates the angular behavior of the dependent variables (z)over time.

FIGS. 14A-14B show (a) Setup for testing of an embodiment of the presentinvention's running gait: 1) an embodiment of the present invention, 2)Higher level controller, 3) LiPo batteries in series, 4) Emergency stop,5) Treadmill. (b) Illustration of trajectory tracking experimentalresults.

FIG. 15 provides a sequence of snapshots showing an embodiment of thepresent invention's running Cycle at 3.2 m/s.

DETAILED DESCRIPTION OF THE INVENTION

Detailed embodiments of the present invention are disclosed herein;however, it is to be understood that the disclosed embodiments aremerely exemplary of the invention, which may be embodied in variousforms. Therefore, specific structural and functional details disclosedherein are not to be interpreted as limiting, but merely as arepresentative basis for teaching one skilled in the art to variouslyemploy the present invention in virtually any appropriately detailedmethod, structure or system. Further, the terms and phrases used hereinare not intended to be limiting, but rather to provide an understandabledescription of the invention.

In one embodiment, the present invention, as shown in FIG. 1, representsan advancement to simplify dynamic legged locomotion while preservingthe characteristics of articulated legs. Provided are leg 100 comprisingdriving disk 110, HFE link 120, femur 130, KFE link 140, tibia 150,double compliance (spring and rubber foot tip) 160, absolute encoder169. FIG. 1B shows two kinematic loops 170 and 172 used in themechanism. FIG. 1C shows how angular relationships simplify thestructure so that it can be described as an OKC for kinematic purposes.

Hardware Design

The mechanical design of an embodiment of the present invention may bethat of a monopod that may be roughly the size of an average domesticdog's leg, at 0.53 m in height. It weighs approximately 6.1 kg and isconstructed with aerospace-grade Al 6061. It has only one actuateddegree of freedom, driven by a BLDC motor (MOOG BN34-25EU-02LH) with 355W power, 2.19 Nm peak torque, and 0.66 Nm continuous torque that ismounted with a 2 stage, 32:1 planetary gearbox and is placed behind thedriving link. An absolute encoder (US Digital MA3-A10-125-B), may bemounted at a distance and connected to the driving disk via a timingbelt to obtain position feedback. An incremental encoder (US DigitalE2-5000-315-IE-H-G-3) is mounted for velocity feedback at the back ofthe motor. Finally, two-stage compliance is provided in the form of aspring and a rubber pad at the foot to withstand the impact during thestance phase.

In one embodiment, the present invention provides a CKC mechanism withtwo closed loops. Loop 1 is comprised of a four-bar mechanism withpassive joints at A, B, and C as marked in FIG. 2. As shown in FIG. 1A,it may comprise a femur 140, which is guided by the driving disk 110with the help of the hip flexion-extension (HFE) link 120. Loop 2 isanother four-bar mechanism that comprises of the knee flexion-extension(KFE) link 150 that couples with the motion of HFE link 130 and guidestibia 160 to execute a smooth continuous trajectory at the foot.

Kinematic Simplification

In other aspects, the present invention provides an embodiment where themechanism eliminates the requirement of using a multi-actuatorcoordination, observed in open kinematic chain mechanisms (OKCs), totake a single step. The intuitive mechanism couples the hip and kneeflexion/extension and thus requires only one actuator for whom, a singlerevolution corresponds to a single stride. Furthermore, this approacheliminates a layer of kinematic computation. Through kinematic loopclosure equations, the system can be represented as an OKC, as seen inFIG. 1A. The arrangement of the linkages simplifies the inversekinematic control problem merely to a one-to-one mapping. The positionof the foot, f, is directly linked to the absolute angle of the motor,θ₁. It is then obvious that at any position of the input crank given bythe absolute encoder, the position of the foot is known to the systemwithout any significant calculations.

The one-to-one mapping can be implemented in the controller through alookup table, vastly reducing the computational requirements by removingthe need for a foot trajectory generator and the calculation of inversekinematics.

Parametric Optimization of the Mechanism

Conventional legs built for dynamic locomotion have access to the 3Dworkspace, and in certain cases, only the 2D workspace, as dictated bythe number of actuators provided per leg. This allows for variousgaits/maneuvers and on the fly adjustments. However, the single degreeof freedom approach significantly curtails the workspace and restrictsthe foot to a single traceable trajectory. Therefore, careful design ofthe mechanism is required to achieve the desired performance.

In other embodiments of the present invention, a six-bar mechanism maybe integrated onto a quadrupedal platform to perform movements such astrot/trot-running gait. As a result, the present invention, as shown inFIGS. 2A and 2B, provides a desired flight-phase trajectory. In apreferred embodiment, the stance trajectory should be sinusoidal.

The combined trajectory can be abstracted by a polynomial, f_(des)(θ₁),where θ₁ is the angle made by the crankshaft with the x-axis of thereference frame at point D. Note that the domain of θ₁∈[0 360], and forall computations in this work, counterclockwise is considered positive.The mechanism is illustrated again in FIG. 2B which illustrates criticalpivot locations A=[A_(x)A_(y)], B=[B_(x)B_(y)], C=[C_(x)C_(y)], andG=[G_(x)G_(y)] that affect the trajectory directly. The leg isparameterized in FIG. 7 and the same convention may be followed. Thedesign space is then described by the set {l_(c) l₁ l₂ l₃ l₄ l₅ l_(s)l_(f) l_(t) θ}. An extra parameter of interest is the angle of attack,a, that the leg makes with the vertical during impact with the ground.The desired trajectory and the minimal angle of attack lead to theformulation of the multi-objective function in Eq. 1, with weights W₁and W₂.

$\begin{matrix}{{W_{1}{\sum\limits_{\theta_{1} = {85{^\circ}}}^{100{^\circ}}{\min\;{f(\alpha)}}}} + {W_{2}{\sum\limits_{\theta_{1} = {0{^\circ}}}^{360{^\circ}}\left( {{f\left( \theta_{1} \right)} - {f_{des}\left( \theta_{1} \right)}} \right)^{2}}}} & (1)\end{matrix}$

Here f (α) is straight-forward, and f (θ₁) is the current position ofthe foot with respect to the crankshaft angle θ₁.

Optimization Results

The optimization results in link lengths, and angle that generate atrajectory, which closely traces the desired trajectory as shown in FIG.4B. The angles corresponding to the first loop are depicted in FIG. 3,due to their importance to the knee trajectory.

While not pictured, the second loop angles undergo a similar evolution.The optimization's validity is proven in the result of the first loop,where the trajectory of the knee is shown to be constrained in the firstquadrant, as compared to the trajectory that was the result of heuristiclink lengths, as in FIG. 4A.

The second loop may then built upon this outcome, resulting in asinusoidal stance phase. The flight phase trajectory in FIG. 4B can beseen to possess a slight concavity near the apex, which is attributed tosatisfying the multiple inequality constraints posed to the system.However, the effect of this concavity on the gait is trivial, as thetrajectory still possesses a rapid entrance into the stance phase.

Dynamic Model of Present Invention

This section develops a mathematical description that leverages SPFformulation for imposition onto the hybrid dynamics framework toalleviate modeling difficulties. The general dynamic equation of motion(EOM) of a system with n links in independent generalized coordinates,denoted by the vector q∈

^(n) ^(q) formulated as:H(q){umlaut over (q)}+C(q,{dot over (q)}){dot over (q)}+g(q)=Bτ+F_(ext)  (7)

where H (q)∈

^(n) ^(q) ^(×n) ^(q) represents the generalized mass matrix, C(q,{dotover (q)})∈

^(n) ^(q) contains the Coriolis and centrifugal terms, is thegravitational term, B(q) is the torque distribution matrix, and τ∈

^(n) ^(q) is the torque vector provided by the actuators. Equation 7results in second-order ODEs identical in dimension to the number ofdegrees of freedom of the system. This implies that for each DOF thereis an independent control input. Furthermore, this formulation possessesseveral structural properties that are favorable in control design.However, CKCs like the present invention do not enjoy the sameproperties, as they are identified by high-index DAEs. The implicitnature of DAEs suggests the presence of dependent variables in thedynamic model that describes the mechanism. This coupling of dependentvariables with independent variables impedes control design, and hencesuch a model is not desired. Most importantly for the development ofcontrol algorithms, as highlighted above, it is desired for the model tobe defined only by the independent variables

Hybrid Dynamics Framework

In other aspects, the present invention provides a hybrid dynamic model.For the hybrid system, a 4-tuple

=(

, S, Δ,

) may be utilized.

is a set of two domains, where

_(S) is the stance domain, and

_(f) is the flight domain. The stance domain is where the leg is incontact with the ground, and the flight domain is where the leg is inthe aerial phase. Both domains represent continuous dynamics but differdue to the addition of two coordinates in the flight phase that map theposition of the center of mass with respect to the inertial frame, {O}as seen in FIG. 7. S:={S_(S), S_(f)} is a set of guards that encodes thestate of the robot at a transition from

_(S) to

_(f) and vice-versa. Note that the constituents of set S are discreteevents. The continuous and discrete dynamics are tied together by Δ, aset of switching functions. Δ feeds appropriate initializations for thecorresponding field,

, of continuous dynamics. A visual representation tailored forembodiments of the present inventions during sagittal running isillustrated in FIG. 5. The mathematical description of each modefollows.

Underlying Constrained EOMs (Flight Phase Dynamics)

The DAE is first established followed by ODE approximation for the“unpinned system” in-flight phase. The method of virtual separation isadapted to derive the dynamic model of the CKC mechanism underconsideration. First, this method prescribes a separation of joints atstrategic locations to form serial and branched kinematic chains asshown in FIG. 6.

Such a system may be denoted as an “unconstrained system.” Traditionalmethods used for serial chains can then be applied to formulate theunconstrained system's EOMs. To capture the dynamic configuration ofthis floating base system, two coordinate frames are defined, aninertial reference frame {O} and a body-fixed frame {B}. In the flightphase, two extra coordinates, x_(b), and y_(b), are added to track theposition of the body with respect to {O}. The absolute orientation ofthe monopod in the sagittal plane is notated in q_(Pitch).

In addition, each link's configuration relative to its previous frame isrepresented by q_(i), with i={1, . . . , 5}. These variables arecollected in the vector q_(d):=[q₁q_(Pitch)x_(b)y_(b)q₂q₃q₄q₅]^(T) andare illustrated in FIG. 2. Further, the masses of the bodies are givenby m_(i), and the respective inertias are etched in I_(i). Similarly,the link lengths are denoted by l_(i), and the position of the center ofmass of each body is given by l_(cm,i). With the help of the Lagrangianformulation, the EOMs of the unconstrained system are then given by:H′(q _(d)){umlaut over (q)}+C′(q _(d){dot over (q)}_(d)){dot over(q)}_(d) +g′(q _(d))=Bτ+F _(ext)  (8)

Here, H′(q_(d))∈

^(8×8), C′(q_(d)) ∈

^(8×8) and g′(q_(d))∈

⁸.

For the sake of brevity, the elements on the right-hand side of theequation, τ and F_(ext), are dropped. This term is not affected by thedefined process and can be added back later without any effort.

Next, this method dictates the incorporation of constraint equationsgiven by ϕ (q_(d)) into the mathematical description of the system, thusreconnecting the separated joints and resulting in a constrained system.The corresponding constraint definitions are provided in FIG. 7. Theresultant system in the flight phase is characterized by constrainedEOMs that are DAEs and assume the form of (9).

$\begin{matrix}\left\{ \begin{matrix}{{{H^{\prime}\left( q_{d} \right)}\overset{¨}{q}} + {{C^{\prime}\left( {q_{d},{\overset{.}{q}}_{d}} \right)}{\overset{.}{q}}_{d}} + {g^{\prime}\left( q_{d} \right)}} \\{{\phi\left( q_{d} \right)} = 0}\end{matrix} \right. & (9)\end{matrix}$

SPF Dynamic Formulation

In other aspects, an embodiment of the present invention may be designedto completely avoid handling the DAEs by approximating them asequivalent ODEs. Due to the kinematic coupling present in the mechanism,q₁ alone is sufficient to describe the leg's motion, which in thetraditional sense implies that a single ODE is sufficient tocharacterize the dynamics of the system. Since it is a planar floatingbase system, the additional three coordinates x, y, and q_(Pitch) arenecessary for a complete description. These four coordinates aretherefore termed as independent variables and are collected in thevector, q:=[q₁q_(Pitch)x_(b)y_(b)]^(T). The surplus variables in q_(d)are the dependent variables and are collected in a separate vector,z:=[q₂q₃q₄q₅]^(T). To eliminate the first order derivative terms of z in(3) and obtain an explicit description of the CKC monopod, thesingularly perturbed dynamic model for fixed base models is leveragedfor this floating base dynamic model.

Given that this minimal order model revolves around the representationof DAEs as ODEs, the problem hinges upon the approximation of thealgebraic constraints. Therefore, a variable w:=ϕ(q_(d)) is introducedto capture the degree of constraint violation. Ideally, it is desiredfor this value to asymptotically converge to zero. By definition, w isan arbitrary variable, allowing the flexibility to decide its dynamicbehavior. Hence, we designate {dot over (w)}=−1/ε*w to assureconvergence to the invariant set {0}. Here, ε can accommodate any smallpositive number. By definition of w, this relationship can then berewritten as in Eq. 10

$\begin{matrix}{{{J_{z}\overset{.}{z}} + {J_{q}\overset{.}{q}}} = {{- \frac{1}{ɛ}}{\phi\left( q_{d} \right)}}} & (10)\end{matrix}$

where J_(z) and J_(q) are the Jacobian matrices. Note that the inclusionof Eq. 9 introduces “fast dynamics” into the model, thus eliminating thealgebraic equations. However, the governing ODE in Eq. 9 is stillcoupled with the second order terms of the dependent variables in z.Therefore, a dimensionality reduction process is undertaken. To begin,we will consider two selector matrices S_(q) and S_(z) to encapsulatethe relationship that q and z hold with q_(d). This correlation can bedenoted as [q z]^(T)=[S_(q) S_(z)]^(T)q_(d). Then, Γ(q_(d)) is formed bycombining ϕ(q_(d)) and S_(q)(q_(d)) as in Eq. 11.

$\begin{matrix}{{\Gamma\left( q_{d} \right)}:=\begin{bmatrix}{\phi\left( q_{d} \right)} \\{S_{q}\left( q_{d} \right)}\end{bmatrix}} & (11)\end{matrix}$

Additionally, we can define ({dot over (q)}_(d))=ρ(q_(d)){dot over (q)}.From this definition, ρ is then given as:

$\begin{matrix}{{\rho\left( q_{d} \right)} = {{\Gamma^{- 1}\left( q_{d} \right)}\begin{bmatrix}0 \\I_{n_{q} \times n_{q}}\end{bmatrix}}} & (12)\end{matrix}$

With this, the dimensionality reduction can then be performed by notingEq. 12. The reduction can be verified by observing the real coordinatespaces: H(q_(d)) ∈

^(4×4), and C(q_(d)) ∈

^(4×4) and g(q_(d))∈

⁴. Finally, the model can be pieced together as in Eq. 14, by replacingq_(d) with (q,z) and combining Eq. 10, Eq. 13, and the torque terms asin Eq. 7.

                                           (13)$\left\{ {{\begin{matrix}{{{H^{\prime}\left( q_{d} \right)}\overset{¨}{q}} + {{C^{\prime}\left( {q_{d},{\overset{.}{q}}_{d}} \right)}{\overset{.}{q}}_{d}} +} \\{g^{\prime}\left( q_{d} \right)}\end{matrix}❘{\begin{matrix}{{H\left( {q,z} \right)} = {{\rho\left( q_{d} \right)}^{T}{H^{\prime}\left( q_{d} \right)}}} \\\begin{matrix}{{C\left( {q_{d},{\overset{.}{q}}_{d}} \right)} = {{{\rho\left( q_{d} \right)}^{T}{C^{\prime}\left( {q_{d},{\overset{.}{q}}_{d}} \right)}{\rho\left( q_{d} \right)}} +}} \\{{\rho\left( q_{d} \right)}^{T}{H^{\prime}\left( q_{d} \right)}{\overset{.}{\rho}\left( {q_{d},{\overset{.}{q}}_{d}} \right)}}\end{matrix} \\{{g\left( q_{d} \right)} = {{\rho\left( q_{d} \right)}^{T}{g^{\prime}\left( q_{d} \right)}}}\end{matrix}\mspace{751mu}{{(14)\mspace{79mu}\begin{bmatrix}{H\left( {q,z} \right)} & 0_{n_{q} \times 1} \\0_{n_{z} \times 1} & {J_{z}\left( {q,z} \right)}\end{bmatrix}}\begin{bmatrix}\overset{¨}{q} \\\overset{.}{z}\end{bmatrix}}}} = \begin{bmatrix}{{{- {C\left( {q,\overset{.}{q},z,\overset{.}{z}} \right)}}\overset{.}{q}} - {g\left( {q,z} \right)} + {B\;\tau} -} \\{{\frac{1}{ɛ}{\phi\left( {q,z} \right)}} - {{J_{q}\left( {q,z} \right)}\overset{.}{q}}}\end{bmatrix}} \right.$

Eq. 14 is the ODE approximation representing the dynamics of the PRESENTINVENTION's flight domain,

_(f), which is visualized in FIG. 8.

Impact Model/Reset Map

The impact model is incorporated in the reset map from flight to stancephase and is Δ_(f) ^(s). General assumptions are made to arrive at thisimpact map. This map resets the initial conditions going into the stancephase, hence the name reset map. It assumes that pre-impact states,(q_(f) ⁻, {dot over (q)}_(f) ⁻), from the flight-phase dynamics areaccessible. Post impact states, (q_(s) ⁺, {dot over (q)}_(s) ⁺), arethen provided as an output. Here, the collision is assumed instantaneousand is modeled as an inelastic collision. This implies that the positionof the feet pre-impact denoted by q⁻, and the position of the feet afterimpact represented by q⁺, are invariable, i.e., q⁻=q⁺. Furthermore, animportant assumption is that there is no slippage between the feet andground on collision is made. The impact map, Eq. 15 is solved for {dotover (q)}⁺, the generalized velocity after impact.H(q ⁺){dot over (q)}⁺ −H(q ⁻){dot over (q)}⁻ =F _(ext)  (15)

Likewise, {dot over (q)}⁻ is the velocity prior to impact. Here, theexternal force, F_(ext), at the foot end is derived through theprinciple of virtual work and is projected onto the joint space as:F _(ext) =J _(c)(q,z)^(T) F  (16)

Where,

${J_{c}\left( {q,z} \right)} = \frac{\partial{p\left( {q,z} \right)}}{\partial q}$

is the Jacobian of the foot position with respect to {O} andF=[F_(T)F_(N)]^(T) is the vector of tangential and normal forces at thefoot end.

Monopod Running Simulation

In the absence of a closed form solution to the dynamics of the hybridnon-linear system, the SPF-hybrid dynamic model of the CKC derived abovemay be validated through numerical simulation. In order to focus thesimulation on the verification of the SPF framework and to replicateconstraints on the experimental setup, the analysis is restricted to thesagittal plane. Furthermore, q_(Pitch) is equated to zero.

Simulation Implementation

The simulation is initialized from the flight phase and is fed with a14-dimensional initial value vector. The initial conditions include thedependent velocities, abstracted as ż. However, the output of the SPFhybrid dynamic model then reduces the system to a 10-dimensional outputthrough the decay of the SPF fast dynamics. Upon impact, these outputsare fed to the reset map, and the stance phase initial conditions arecalculated. In the stance phase, the fixed frame position and velocitycan be extracted using the relationship between the foot and the fixedframe, as the foot is considered a pivot point during this phase.

Once the desired phase angle is reached, a predetermined set point forthe angle between the foot and the body fixed frame at the hip as seenin FIG. 7, the transition to flight phase is triggered. The stance phasestates are multiplied with the identity matrix encoded within the flightmap in order to provide the initial conditions to map back into theflight phase. This cycle is repeated for each step.

It becomes clear that some form of control is necessary in order to takea single step. However, the focus is to show the validity of the SPFmodel, we seek a simple controller. For monopod running, to move the legto the desired angle of attack, α_(m) ^(des), before the next impact isthe most basic-level control requirement. The control law is specifiedin Eq. 17.τ=K _(p)(θ₁ ^(des)(α_(m))−θ₁)−K _(D){dot over (θ)}₁  (17)

Here, θ₁ is the measured angle of the crankshaft, {dot over (θ)}₁ is themeasured angular rate, and K_(P) and K_(D) are the proportional andderivative gains, respectively.

Results

The simulation was performed for 12 steps, and frames of the simulationduring flight-phase are shown in FIG. 9. The simulation demonstrated theSPF model was successful in the hybrid framework, as shown by theconstraints holding throughout the simulation. The constraint errorswere abstracted by w₁-w₄, where w₁ and w₂ correspond to the x and yerrors of the constraint equation generated by the condition e₀=e₁,while w₃ and w₄ represent the x and y errors for the condition g₀=g₁.

As seen in FIG. 10, constraint errors asymptotically track zero andmaintain this behavior during state transitions. In the figure, blocks400-407 highlight the transition from stance to stride phase and viceversa, while blocks 420-423 denote the stance phase.

FIG. 10 shows the response of q₁ during the simulation. It can be seenthat q₁ shows a periodic limit cycle. Note that the angle does notprescribe a full rotation due to the simplicity of the controllerutilized in the simulation, but rather oscillates within a short range.However, this does not prevent the leg from following a satisfactorytrajectory, it is simply inefficient to implement in actuality. Theresults of the implicit angles are found in FIG. 13. The ranges shown bythese angles correspond to the ranges obtained in the optimization,relative to the motion of q₁. The sharp drops in the angular plots andthe limit cycles signify impact.

FIG. 11A shows the x and y position of the body-fixed frame with respectto the inertial frame over time. This demonstrates that the simulationsuccessfully obtained monopod running, with a monotonically increasingx-position and a cyclical y-position. The periodicity in y and q₁ areshown in FIG. 11B through the limit cycle. In these, the verticalstraight lines indicate the instantaneous impact.

The pitch angle was restricted as it is unnecessary to demonstrate thestability of the system as it is inherently stable.

Setup

In certain aspects, when the present invention is planar, its mobilityis constrained to the sagittal plane using a custom framing, as shown inFIG. 14A. The framing may be mounted onto a commercially availabletreadmill to evaluate the running performance of the robot. The motor iscommanded by the low-level controller via a Teensy microcontroller,while a higher-level controller runs on a computer and provides theTeensy with speed commands. Three 14.8 V LiPo batteries are connected inseries (32.56 Whrs) to power the robot.

A first focus was on a trajectory validation, where the leg was raisedabove the treadmill surface and is constrained in both the x and ydirection. A visual object tracking system, LOSA, was attached to therubber pad of the foot. The device was then run for a set period, andthe trajectory of the foot was recorded.

Another focus was to demonstrate open-loop running, wherein the leg wasunconstrained in the x and y directions. A minimum y position wasimposed with a bumper to protect the hardware. The device was run atmultiple speeds from 0.5 m/s to a maximum of 3.2 m/s to observeconsistency of performance, with the treadmill speed matched in order toachieve in-place running.

Results

Leveraging the millimeter accuracy of the LOSA object tracking system,the foot trajectory was recorded and is shown in FIG. 14B. The resultsreveal the success of the optimization, as the stride length and heightcorrespond to those resulting from the kinematic simulations performedfollowing the optimization. As evident from FIG. 14B, the foot tracked atrajectory close to the designed trajectory. The small inconsistenciesin the trajectory are due to vibration in the framing due to highspeeds. Open loop running can be seen in the sequence shown in FIG. 15.The sequence depicts a single stride cycle at 3.2 m/s. The open-looprunning shows the promise of the mechanism in performing quadrupedalgaits on flat terrain, as well as its capability to achieve the targetedspeed.

While the foregoing written description enables one of ordinary skill tomake and use what is considered presently to be the best mode thereof,those of ordinary skill will understand and appreciate the existence ofvariations, combinations, and equivalents of the specific embodiment,method, and examples herein. The disclosure should therefore not belimited by the above-described embodiments, methods, and examples, butby all embodiments and methods within the scope and spirit of thedisclosure.

What is claimed is:
 1. A legged robot with two closed kinematic loops,comprising: a vertically oriented floating-base link, said floating-baselink having one active joint and one passive revolute joint arranged,parallel to a frontal plane and having a driving link connected to saidactive joint; a hip flexion and extension link connected to an end ofsaid driving link with a second revolute joint; a knee flexion andextension link mounted on said second revolute joint adjacent to saidhip flexion and extension link, adjacent to a point of attachment withthe said driving link; a femur link mounted on said passive revolutejoint, said femur link having two other revolute joints, one adjacent toa point of attachment with said floating-base link and one at the otherend on which a tibia link is mounted; a free end of said hip flexion andextension length is mounted on said first revolute joint of said femurlink adjacent to the point of attachment with said floating-base linkforming the first closed kinematic loop; said tibia link has a revolutejoint located adjacent to a connecting point with said femur link, whichconnects said free end of said knee flexion and extension link, formingsaid second closed kinematic loop; and a distal end of the tibia link ismounted with a compression spring of predetermined stiffness value toprovide compliance during impact.
 2. The legged robot of claim 1,wherein a lower end of the spring is fitted with a compliant rubber footto provide a second-stage of compliance against impact forces.
 3. Thelegged robot of claimed claim 2, wherein a flight-phase trajectory ofthe foot is jerk free and has a retraction rate that reduces energylosses at touchdown.
 4. The legged robot of claim 1 wherein only oneactuator is used to perform continuous dynamic locomotion.
 5. A dynamicquadrupedal robot comprising: four legs; each leg having a verticallyoriented floating-base link, with one active joint and one passiverevolute joint arranged, parallel to a frontal plane and having adriving link connected to said active joint; a hip flexion and extensionlink connected to the other end of the driving link with anotherrevolute joint; a knee flexion and extension link mounted on a revolutejoint located on said hip flexion and extension link, adjacent to apoint of attachment with said driving link; a femur link mounted on saidpassive revolute joint, said femur link having two other revolutejoints, one adjacent to a point of attachment with said floating-baselink and one at the other end on which a tibia link is mounted; a freeend of said hip flexion and extension length is mounted on said firstrevolute joint of said femur link adjacent to the point of attachmentwith said floating-base link, forming a first closed kinematic loop;said tibia link has a revolute joint, located adjacent to a connectingpoint with said femur link, which connects a free end of the said kneeflexion and extension link, forming a second closed kinematic loop; anda distal end of the tibia joint is mounted with a compression spring ofpredetermined stiffness value to provide compliance during impact. 6.The quadrupedal robot of claim 5 wherein the front legs and back legsare mounted on two different sagittal planes creating an offset forenhanced stability.